Quantum Country exercise 10

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 10. Show that $M \vert e_k \rangle$ is the $k$ th column of the matrix $M$ and that $\langle e_j \vert M \vert e_k \rangle$ is the $jk$ th entry of $M$ .

Answer: The product of the matrix $M$ and the vector $\vert e_k \rangle$ is a vector, with the $j$ th entry of that vector being the inner product of the $j$ th row of $M$ with $\vert e_k \rangle$ :

$\left( M \vert e_k \rangle \right)_j = \sum_l M_{jl} \vert e_k \rangle_l$

We then have

$\sum_l M_{jl} \vert e_k \rangle_l = M_{jk} \vert e_k \rangle_k + \sum_{l \ne k} M_{jl} \vert e_k \rangle_l$

But by the definition of $\vert e_k \rangle$ we have $\vert e_k \rangle_k = 1$ and $\vert e_k \rangle_l = 0$ for $l \ne k$ . So the above reduces to

$M_{jk} \vert e_k \rangle_k + \sum_{l \ne k} M_{jl} \vert e_k \rangle_l = M_{jk} \cdot 1 + \sum_{l \ne k} M_{lj} \cdot 0 = M_{jk}$

So $\left( M \vert e_k \rangle \right)_j = M_{jk}$ for all $j$ , which means that $M \vert e_k \rangle$ is the $k$ th column of $M$ .

We now consider the product $\langle e_j \vert M \vert e_k \rangle$ for some $j$ . This is the inner product of the vector $\langle e_j \vert$ with the vector $\vert M \vert e_k \rangle$ , which we have just shown is the $k$ th column of $M$ . We therefore have